What is the total number of ways of choosing $4$ cards from a pack of $52$ playing cards? In how many of these are the $4$ cards of the same colour?

If the number of $5$-element subsets of the set $A = \{a_1, a_2, \dots, a_{20}\}$ containing $20$ distinct elements is $k$ times the number of $5$-element subsets containing the element $a_4$,then what is the value of $k$?

An examiner can assign $30$ marks to $8$ questions. If he does not assign less than $2$ marks to any question,in how many ways can he assign the marks?

$A$ committee of $7$ members is to be formed from $9$ boys and $4$ girls. In how many ways can this be done if the committee must consist of exactly $3$ girls?

Find the number of ways of selecting $4$ pens and $3$ pencils from a packet of $8$ pens and $5$ pencils.

To fill $12$ vacancies,there are $25$ candidates,of which $5$ are from the scheduled caste. If $3$ of the vacancies are reserved for scheduled caste candidates while the rest are open to all,then the number of ways in which the selection can be made is: